with a nice shortcut and an interesting answer: Q: A cylindrical water tank, radius 1, rests on its side, skewered by a coaxial bar of radius r. Choose r to maximize capacity if the water level may not exceed the top of the bar. A: .retal etaitnereffid od gniog tsuj er'uoy ecnis )2^x-1(trqs largetni rotces eht od ot deen t'nod uoy taht si tuctrohs ehT .)2^ip+1(trqS/1